Derive expressions for (∂u / ∂P)T and (∂h / ∂∪)T in terms of P, U, and T only.

Name: Problem 51, Chapter 12: Thermodynamics: An Engineering Approach
Uploaded: 2022-01-28T11:39:56-08:00
Duration: 6 min 25 s
Channel: Mohammad Mehran
Description: Derive expressions for (∂u / ∂P)T and (∂h / ∂∪)T in terms of P, U, and T only.

Derive expressions for (∂u / ∂P)T and (∂h / ∂∪)T in terms of...

Key Concepts

Enthalpy Definition

Enthalpy is defined by the relation h = u + Pv, which combines the system’s internal energy with the work done by or against its surroundings at constant pressure. This definition links the concepts of energy storage and work and allows subsequent differentiation to yield useful relationships under constant temperature or pressure conditions.

Legendre Transformations

Legendre transformations are used to change the natural variables of a thermodynamic potential. For instance, while the internal energy u is a function of entropy and volume, the enthalpy h is obtained via a Legendre transform to be a function of entropy and pressure. This change simplifies the treatment of systems under constant pressure and is instrumental in deriving expressions that involve P, U, and T only.

Thermodynamic Potentials

Thermodynamic potentials such as internal energy (u) and enthalpy (h) are state functions that characterize the energy of a system under different conditions. The internal energy is naturally expressed as a function of entropy and volume, while enthalpy, defined as h = u + Pv, is particularly useful for processes occurring at constant pressure. Understanding these potentials provides the groundwork for deriving relationships among thermodynamic variables.

Chain Rule and Partial Differentiation

In thermodynamics many state functions depend on several independent variables. The chain rule allows one to relate derivatives with respect to different variables when some variables are held constant. By expressing a derivative with respect to one variable via an intermediate dependence (for example, writing (?u/?P)_T as a product of (?u/?V)_T and (?V/?P)_T), one can reframe complex derivatives in terms of measurable quantities.

Maxwell Relations

Maxwell relations derive from the equality of mixed second partial derivatives of thermodynamic potentials. They relate derivatives of entropy with respect to other variables (such as volume or pressure) to derivatives of measurable quantities like pressure with respect to temperature. These relations are key to expressing derivatives of u or h with respect to pressure in terms of other variables such as T and P.

Transcript

00:01 All right, guys, to find the relation for partial derivative u by partial derivative p with respect to t, we will need to use the cyclic relation to determine the partial derivative, partial derivative p over partial derivative t with respect to u multiplied by partial derivative v divided by portion derivative p with respect to t so now we solve it partial derivative p divided by partial derivative t with respect to v multiplied by partial derivative t divided by partial derivative v with respect to p so partial derivative v divided by partial derivative p with respect to t is equal to minus one so guys minus partial derivative p divided by partial derivative t with respect to v into partial derivative v over partial derivative p with respect to t is equal to minus partial derivative v divided by partial derivative t with respect to p next we can use the general relation to for derivative of d of u and differentiate it with respect to p under constant t so derivative of u is equal to c v d t plus t into partial derivative p divided by partial derivative of t minus p multiplied by derivative of v so partial derivative u divided by partial derivative p with respect to t is equal to t into partial derivative b divided by partial derivative t minus p so we multiplied it by partial derivative v divided by partial derivative b b divided by partial derivative b and solving it further we get partial derivative u divided by partial derivative p with respect to t is equal to minus t and multiplied by partial derivative b over partial derivative t with respect to p minus p into partial derivative v over partial derivative b with respect to t now to find the relationship for partial derivative h over partial derivative b with respect to t we need to use the cyclic relation to determine so we will use partial partial derivative, okay, we will use the partial derivative v over partial derivative t with respect to p, multiply by partial derivative t divided by partial derivative p with respect to v into partial derivative p divided by partial derivative v is equal to minus 1.

04:07 So now taking it on the other side, after rearranging, we get partial derivative v, partial derivative divided by partial derivative t with respect to p, and partial derivative p divided by partial derivative b with respect to t is equal to minus partial derivative t and partial derivative p with respect to v.

04:38 Next, we can use the general relation for derivative of h and differentiated with respect to v under t.

04:48 So dh is equal to cp dt plus v minus t into partial derivative of b and divided by partial derivative of t with respect to p into derivative of p...

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